Planar Planar and Non-planar graphsand Non-planar graphs
- 1. Planar and Non-planar graphs Presented by: Supervisor: Snarya S. Hussein Dr. Didar A. Ali Fatma M. Ahmed
- 2. Outline • Definition • Note • Example • Theorem (Euler’s Formula) • Example • Theorem • Corollary • Example • Definition • References
- 3. Definition • A graph is planar if it can be drawn so that its edges do not cross.
- 4. Note 1. The Path Graph of order n is a planar graph for all 𝑛 2. The Cycle Graph of order n is a planar graph for all 𝑛 3. The wheel Graph of order n is a planar graph for all 𝑛 4. The star Graph of order n is a planar graph for all 𝑛 5. The complete Graph of order n is a planar graph for all 𝑛 ≤ 4 The complete bipartite graph is planar if either 𝑚 = 1 or 𝑛 = 1 also 𝐾2,2. The graph 𝐾3,3 is a non-planar graph.
- 5. Definition • Bounded regions defined by planar graphs are called faces (regions), but unbound region called exterior face.
- 6. Example This graph has a total of three faces: f = 3 Two interior faces (A, B) One exterior face (C)
- 7. Theorem: Euler Formula If 𝐺 is a connected plane graph, then 𝑉 − 𝐸 + 𝐹 = 2. In the above theorem or formula, |𝑉|, |𝐸|, and |𝐹| denote the number of vertices, edges, and faces of the graph G respectively.
- 8. A connected planar graph has 5 vertices and 7 edges, How many faces dose the graph have? Solution: Euler’s Formula 𝑉 − 𝐸 + 𝐹 = 2 5 − 7 + 𝐹 = 2 𝐹 = 2 + 2 𝐹 = 4 So number of faces is 4. Example.
- 9. Theorem: For any (𝑝, 𝑞) planar graph in which every face is 𝐶𝑛 then, 𝑞 = 𝑛(𝑝−2) n−2 Example: Does there exists a planar graph of order 18, and each face is 𝐶5, if exists construct it, and then find the number of faces. Solution: If the graph exists, then must satisfy the equation. 𝑞 = 5(18 − 2) 5 − 2 = 26.666 Which is not integer, therefore the graph does not exists.
- 10. Corollary If G is any (𝑝, 𝑞) planar graph with 𝑝 ≥ 3, then 𝑞 ≤ 3𝑝 − 6, if G has no triangle, then 𝑞 ≤ 2𝑝 – 4
- 11. Is K5 is a planar graph? Solution: if K5 is planar graph, then must satisfies the inequality 𝑞 ≤ 3𝑝 − 6 𝑞 𝐾5 = 5(4) 2 = 10 𝑎𝑛𝑑 𝑝 = 5 ∴ 10 ≤ 3 5 − 6 ⇒ 10 ≤ 15 − 6 10 ≤ 9 Sine 10 is not less or equal to 9 ∴ 𝐾5 is not planar graph. Example 1.
- 12. Is K3,3 is a planar graph? Solution: is K3,3 is a planar graph then must satisfies the inequality 𝑞 ≤ 2𝑝 − 4 𝑞 𝐾3,3 = 3 × 3 = 9 𝑎𝑛𝑑 𝑝 = 3 + 3 = 6 ∴ 9 ≤ 2 6 − 4 ⇒ 9 ≤ 12 − 4 9 ≤ 8 sine 9 not less or equal to 8 ∴ 𝐾3,3 is not planar graph Example 2.
- 13. Definition A maximal planar graph is a graph in which no edge can be added without losing planarity. Example: the graphs 𝐾3,3 − 𝑒 and 𝐾5 − 𝑒 are maximal planar graphs.
- 14. References • Tutte, W. T. (1956). A theorem on planar graphs. Transactions of the American Mathematical Society. • Whitney, H. (1933). Planar graphs. In Hassler Whitney Collected Papers. Boston, MA: Birkhäuser Boston. • Biedl, T., Liotta, G., & Montecchiani, F. (2015). On visibility representations of non-planar graphs.